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3. Simple models
Daniel J. Jacob, Atmospheric Chemistry, Harvard University, Spring 2017
Atmospheric evolution of species X is given by the continuity equation
This equation cannot be solved exactly need to construct model (simplified representation of complex system)
[ ] ( [ ])X X X XX E X P L Dt
∂= −∇• + − −
∂U
local change in concentration
with time
transport(flux divergence;U is wind vector)
chemical production and loss(depends on concentrationsof other species)
emissiondeposition
How to design and use a model
Define problem of interest
Design model; make assumptions neededto simplify equations
and make them solvable
Evaluate model with observations
Apply model: make hypotheses, predictions
Improve model, characterize its error
Design observational system to test model
One-box model
Inflow Fin Outflow Fout
XE
EmissionDeposition
D
Chemicalproduction
P L
Chemicalloss
Atmospheric “box”;spatial distribution of X within box is not resolved
out
Atmospheric lifetime: mF L D
τ =+ + Fraction lost by export: out
out
FfF L D
=+ +
Lifetimes add in parallel: export chem dep
1 1 1 1outF L Dm m mτ τ τ τ
= + + = + +
Loss rate constants add in series:export chem dep
1k k k kτ
= = + +
Mass balance equation: sources - sinks in outdm F E P F L Ddt
= = + + − − −∑ ∑
Nitrogen dioxide (NO2) has atmospheric lifetime ~ 1 day:strong gradients away from combustion source regions
Satellite observations of NO2 columns
Carbon monoxide (CO) has atmospheric lifetime ~ 2 months:mixing around latitude bands
Satellite observations
CO2 has atmospheric lifetime ~ 100 years:global mixing
Assimilated observations
On average, only 60% of emitted CO2 remains in the atmosphere – but there is large interannual variability in this fraction
Using a box model to quantify CO2 sinksPg
C y
r-1
Special case: constant source, first-order sink
( ) (0) (1 )kt ktdm SS km m t m e edt k
− −= − ⇒ = + −
Steady state solution (dm/dt = 0)
Initial condition m(0)
Characteristic time τ = 1/k for• reaching steady state• decay of initial condition
If S, k are constant over t >> τ, then dm/dt 0 and m S/k: quasi steady state
TWO-BOX MODELdefines spatial gradient between two domains
m1 m2
F12
F21
Mass balance equations: 11 1 1 1 12 21
dm E P L D F Fdt
= + − − − +
If mass exchange between boxes is first-order:
11 1 1 1 12 1 21 2
dm E P L D k m k mdt
= + − − − +
system of two coupled ODEs (or algebraic equations if system is assumed to be at steady state)
(similar equation for dm2/dt)
Research models solve mass balance equationin 3-D assemblage of gridboxes
Solve mass balance equation for individual gridboxes
A different approach: follow air parcel moving with the wind(puff model)
CX(xo, to)
CX(x, t)
wind
In the moving puff,
XdC E P L Ddt
= + − −
…no transport terms! (they’re implicit in the trajectory)
We can also have the puff dilute with background air (plume model)
CX
CX,bbackground air
,( )Xdilution X X b
dC E P L D k C Cdt
= + − − − −In plume,
Application: transport and evolution of fire plumes
Fire plumes oversouthern California
Sulfur dioxide (SO2) from Aleutian volcano eruption (8/8/2008)
http://www.avo.alaska.edu/
Kasatochi volcano
Altitude: 314 mLatitude: 52.16°NLongitude: 175.51° W
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